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quation.3 , 345 k ?@>15305B 2A5 G8A;0 2708<=> ?@>ABK5 A n. 0: 284=> 87 MB>9 D>@<C;K ?>AB@>5=85 ?>;8=><0 ?> >?@545;5=8N A>?@O65=> A> A;>6=>ABLN ?5@5<=>65=8O A:>1>: 8 3@C??8@>2:>9 <>=><>2 A :><?;5:A=K<8 :>MDD8F85=B0<8. 50;87>20==K9 20@80=B. 5;8G8=0 d ?@>15305B 2A5 G8A;0, O2;ONI85AO 45;8B5;O<8 n. (d)  DC=:F8O Q18CA0. "0:8< >1@07><, <K 87102;O5<AO >B =C64K @01>BK A :><?;5:A=K<8 G8A;0<8.  @57C;LB0B5 @01>BK ?>AB02;5==K5 7040G8 1K;8 @5H5=K, DC=:F88 1K;8 ?@82545=K : A8=B0:A8G5A:><C AB8;N Arageli 8 <>3CB 1KBL 2:;NG5=K 2 181;8>B5:C. <5=0 DC=:F89 <>6=> ?>A<>B@5BL 2 D09;5  HYPERLINK "Source/>10=>2/special_polynoms.hpp" special_polynoms.hpp. A?>;L7C5<0O ;8B5@0BC@0 WolframMathWorld. Cyclotomic Polynomial.  HYPERLINK "http://mathworld.wolfram.com/CyclotomicPolynomial.html" http://mathworld.wolfram.com/CyclotomicPolynomial.html Alglib  A?5F80;L=K5 DC=:F88.  HYPERLINK "http://alglib.sources.ru/specialfunctions/" http://alglib.sources.ru/specialfunctions/ >=0;L4 -. =CB. A:CAAB2> ?@>3@0<<8@>20=8O, B>< 2. >;CG8A;5==K5 0;3>@8B<K, 3-5 8740=85 >8A: @0F8>=0;L=KE 7028A8<>AB59 459AB28B5;L=KE G8A5;. B25BAB25==K9: "C;C1052  >AB0=>2:0 7040G8 50;870F8O 0;3>@8B<0 ?>8A:0 F5;>G8A;5==KE 7028A8<>AB59 4;O 459AB28B5;L=KE G8A5; ?>4 =0720=85< PSLQ. 0= 25:B>@ (x1,...,xn) 459AB28B5;L=KE G8A5;, =C6=> =09B8 B0:>9 25:B>@ (a1,...,an) F5;KE G8A5;, GB> x1*a1+...+xn*an=0. ".:. Arageli =5 ?@54CA<0B@8205B @01>BK A 25I5AB25==K<8 G8A;0<8 (8A?>;L7C5BAO 8E ?@81;865==K5 G8A;0 A ?;020NI59 70?OB>9), B> @025=AB2> <>6=> >1;53G8BL 4> ?@8<5@=>3> @025=AB20. 5>1E>48<> @50;87>20BL MB>B 0;3>@8B< 2 Arageli, =0?8A0BL 4>:C<5=B0F8N 8 :0=>=8G5A:85 ?@8<5@K, 8A?>;L7CNI85 53> DC=:F8>=0;L=>ABL. >7=8:H85 ?@>1;5<K Arageli O2;O5BAO 181;8>B5:>9 B>G=KE 2KG8A;5=89, 0 0;3>@8B< 4065, 5A;8 =0 2E>4 ?>405BAO 25:B>@ @0F8>=0;L=KE G8A5;, 2> 2@5<O A2>59 @01>BK 2KE>48B 2 <=>65AB25 8@@0F8>=0;L=KE G8A5;.  ?@8<5@C 872;5G5=85 :204@0B=>3> :>@=O. "0: :0: 4;O ?@54AB02;5=8O 8@@0F8>=0;L=KE G8A5; 2 181;8>B5:5 8A?>;L7CNBAO G8A;0 A ?;020NI59 B>G:>9, B> =0 :064>9 8B5@0F88 0;3>@8B<0 845B ?>B5@O B>G=>AB8. 0==0O ?@>1;5<0 1K;0 @5H5=0 A;54CNI8< >1@07><: 22545=85< <8=8<0;L=>9 B>G=>AB8, ?@8 >?CA:0=88 =865 :>B>@>9 0;3>@8B< ?@5:@0I05B A2>N @01>BC. 8G=0O ?>;L70 #G0AB85 2 40==>< <8=8-?@>5:B5 1K;> 4;O <5=O >G5=L ?>;57=K< 2 B>< ?;0=5, GB>: O ?>7=0:><8;AO A =>2>9, <>I=>9 181;8>B5:>9 :><?LNB5@=>9 0;351@K, :>B>@0O ?><>65B <=5 2 40;L=59H5< >1CG5=88 =0 D0:C;LB5B5; ?@8>1@5B5=K =5:>B>@K5 =02K:8 ?> MDD5:B82=><C 2708<>459AB28N 8 ?;0=8@>20=8N 2=CB@8 :><0=4K; ?>;CG5= >?KB =0?8A0=8O 4>:C<5=B0F88; 8, :>=5G=> 65 C:@5?;5=85 7=0=89 2 >1;0AB8 !++. A?>;L7C5<0O ;8B5@0BC@0: AB0BLO  HYPERLINK "8B5@0BC@0/pslq-cse.pdf" David H. Bailey Integer relation detection with PSLQ AB0BLO  HYPERLINK "8B5@0BC@0/cpslq.pdf" Ferguson, Bailey, Arno Analysis of PSLQ algorithm "5AB8@>20=85 "5AB8@>20=85 DC=:F89 >10=>20. ! !<O DC=:F88 H81:8><<5=B0@88 @07@01>BG8:01template<typename F,typename P> void lager_polynom(const F& n, P& poly);H81>: =5 >1=0@C65=>2template<typename T,typename F> sparse_polynom<T,F> lager_polynom(const F& n);H81>: =5 >1=0@C65=>3template<typename P> bool is_lager_polynom(const P& poly);H81>: =5 >1=0@C65=>4template<typename F,typename P> void hermite_polynom(const F& n, P& poly);H81>: =5 >1=0@C65=>5template<typename T,typename F> sparse_polynom<T,F> hermite_polynom(const F& n);H81>: =5 >1=0@C65=>6template<typename P> bool is_hermite_polynom (const P& poly);H81>: =5 >1=0@C65=>7template<typename F,typename P> void chebychev1(const F& n, P& poly);H81>: =5 >1=0@C65=>8template<typename T,typename F> sparse_polynom<T,F> chebychev1(const F& n);H81>: =5 >1=0@C65=>9template<typename P> bool is_chebychev1(const P& poly);H81>: =5 >1=0@C65=>10template<typename F,typename P> void cycle_polynom(const F& n, P& poly);H81>: =5 >1=0@C65=>11template<typename L,typename P> bool is_cycle_polynom (const P& poly, L& poly_n);H81>: =5 >1=0@C65=>12template<typename L, typename P> bool is_cycle_polynom (const P& poly);H81>: =5 >1=0@C65=>13template<typename P> bool is_eizenshtein(const P& poly);H81>: =5 >1=0@C65=>14template<typename P> bool is_perron(const P& poly);H81>: =5 >1=0@C65=> "5AB8@>20=85 DC=:F89 @CG8=8=>9  !<O DC=:F88H81:8><<5=B0@88 @07@01>BG8:01double hiBoundOfRoots (const ComplexPolynomial &val); H81:8 =5 >1=0@C65=K.2int polySeparation(const ComplexPolynomial &p, sparse_polynom<sparse_polynom<double> > &phi, sparse_polynom<sparse_polynom<double> > &psi); 5?>=OB=0O >H81:0. > 2@5<O @01>BK 2K;5B05B 58A?@028<0O >H81:0 windowsH81:8 =5 1C4CB 8A?@02;5=K 2 1;8609H55 2@5<O, B.:. =5 8725AB=0 ?@8@>40 >H81:8. "5AB8@>20=85 DC=:F89 "C;C10520  !<O DC=:F88H81:8><<5=B0@88 @07@01>BG8:01vector<rational<>> find_integer_relation (rational<>* x, int n)$C=:F8O 8<55B >G5=L AK@>9 :>4. @8 @07;8G=KE 2E>4=KE 40==KE ?@>8AE>48B 8;8 7028A0=85, 8;8 2K;5B A >H81:>9.7-70 1>;LH>3> >1J5<0 @01>B ?> 8A?@02;5=8N, MB> =5 1C45B 2K?>;=5=> 2 @0<:0E <8=8-?@>5:B0. B>38  E>45 @01>BK =04 40==K< <8=8-?@>5:B>< <K >;CG8;8 =5?;>E>9 >?KB ?@>3@0<<8@>20=8O A?KB0;8 2AN A;>6=>ABL @01>BK 2 :><0=45 8 A45;0;8 A>>B25BAB2CNI85 2K2>4K. >;CG8;8 >?KB =0?8A0=8O <0=C0;>2. >7=0:><8;8AL A 181;8>B5:>9 Arageli  >G5=L ?>;57=K< 8=AB@C<5=B>< 4;O 87CG5=8O 2KG8A;8B5;L=>9 0;351@K.  8B>35 @01>BK 1K;8 =0?8A0=K 8 ?@>B5AB8@>20=K 17 DC=:F89, @0AH8@ONI8E 2>7<>6=>AB8 181;8>B5:8. !AK;:8  HYPERLINK "65=545;L=K5%20>BG5BK" 65=545;L=K5 >BG5BK.  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